Mixed weak estimates of Sawyer type for commutators of singular integrals and related operators

We study mixed weak type inequalities for the commutator $[b,T]$, where $b$ is a BMO function and $T$ is a Calder\'on-Zygmund operator. More precisely, we prove that for every $t>0$ \begin{equation*}%\label{tesis_teo2.2} uv(\{x\in\R^n: |\frac{[b,T](fv)(x)}{v(x)}|>t\})\leq C\int_{\R^n}\phi(\frac{|f(x)|}{t})u(x)v(x)\,dx, \end{equation*} where $\phi(t)=t(1+\log^{+}{t})$, $u\in A_1$ and $v\in A_{\infty}(u)$. Our technique involves the classical Calder\'on-Zygmund decomposition, which allow us to give a direct proof. We use this result to prove an analogous inequality for higher order commutators. We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of $L\log L$ type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight $u$ and a radial function $v$ which is not even locally integrable.